Finite Math Examples

${(72 x^{2} + x)}^{2} - 74 (72 x^{2} + x) + 73 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = 72 x^{2} + x$ . Substitute $u$ for all occurrences of $72 x^{2} + x$ .

$u^{2} - 74 u + 73 = 0$

Step 1.2

Factor $u^{2} - 74 u + 73$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $73$ and whose sum is $- 74$ .

$- 73, - 1$

Step 1.2.2

Write the factored form using these integers.

$(u - 73) (u - 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $72 x^{2} + x$ .

$(72 x^{2} + x - 73) (72 x^{2} + x - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$72 x^{2} + x - 73 = 0$

$72 x^{2} + x - 1 = 0$

Step 3

Set $72 x^{2} + x - 73$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $72 x^{2} + x - 73$ equal to $0$ .

$72 x^{2} + x - 73 = 0$

Step 3.2

Solve $72 x^{2} + x - 73 = 0$ for $x$ .

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Step 3.2.1

Factor by grouping.

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Step 3.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 72 \cdot - 73 = - 5256$ and whose sum is $b = 1$ .

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Step 3.2.1.1.1

Multiply by $1$ .

$72 x^{2} + 1 x - 73 = 0$

Step 3.2.1.1.2

Rewrite $1$ as $- 72$ plus $73$

$72 x^{2} + (- 72 + 73) x - 73 = 0$

Step 3.2.1.1.3

Apply the distributive property.

$72 x^{2} - 72 x + 73 x - 73 = 0$

Step 3.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.1.2.1

Group the first two terms and the last two terms.

$(72 x^{2} - 72 x) + 73 x - 73 = 0$

Step 3.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$72 x (x - 1) + 73 (x - 1) = 0$

Step 3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (72 x + 73) = 0$

Step 3.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$72 x + 73 = 0$

Step 3.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.2.4

Set $72 x + 73$ equal to $0$ and solve for $x$ .

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Step 3.2.4.1

Set $72 x + 73$ equal to $0$ .

$72 x + 73 = 0$

Step 3.2.4.2

Solve $72 x + 73 = 0$ for $x$ .

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Step 3.2.4.2.1

Subtract $73$ from both sides of the equation.

$72 x = - 73$

Step 3.2.4.2.2

Divide each term in $72 x = - 73$ by $72$ and simplify.

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Step 3.2.4.2.2.1

Divide each term in $72 x = - 73$ by $72$ .

$\frac{72 x}{72} = \frac{- 73}{72}$

Step 3.2.4.2.2.2

Simplify the left side.

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Step 3.2.4.2.2.2.1

Cancel the common factor of $72$ .

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Step 3.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{72 x}{72} = \frac{- 73}{72}$

Step 3.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 73}{72}$

Step 3.2.4.2.2.3

Simplify the right side.

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Step 3.2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{73}{72}$

Step 3.2.5

The final solution is all the values that make $(x - 1) (72 x + 73) = 0$ true.

$x = 1, - \frac{73}{72}$

Step 4

Set $72 x^{2} + x - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $72 x^{2} + x - 1$ equal to $0$ .

$72 x^{2} + x - 1 = 0$

Step 4.2

Solve $72 x^{2} + x - 1 = 0$ for $x$ .

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Step 4.2.1

Factor by grouping.

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Step 4.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 72 \cdot - 1 = - 72$ and whose sum is $b = 1$ .

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Step 4.2.1.1.1

Multiply by $1$ .

$72 x^{2} + 1 x - 1 = 0$

Step 4.2.1.1.2

Rewrite $1$ as $- 8$ plus $9$

$72 x^{2} + (- 8 + 9) x - 1 = 0$

Step 4.2.1.1.3

Apply the distributive property.

$72 x^{2} - 8 x + 9 x - 1 = 0$

Step 4.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.1.2.1

Group the first two terms and the last two terms.

$(72 x^{2} - 8 x) + 9 x - 1 = 0$

Step 4.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$8 x (9 x - 1) + 1 (9 x - 1) = 0$

Step 4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $9 x - 1$ .

$(9 x - 1) (8 x + 1) = 0$

Step 4.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$9 x - 1 = 0$

$8 x + 1 = 0$

Step 4.2.3

Set $9 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.2.3.1

Set $9 x - 1$ equal to $0$ .

$9 x - 1 = 0$

Step 4.2.3.2

Solve $9 x - 1 = 0$ for $x$ .

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Step 4.2.3.2.1

Add $1$ to both sides of the equation.

$9 x = 1$

Step 4.2.3.2.2

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 4.2.3.2.2.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.2.3.2.2.2

Simplify the left side.

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Step 4.2.3.2.2.2.1

Cancel the common factor of $9$ .

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Step 4.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 4.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 4.2.4

Set $8 x + 1$ equal to $0$ and solve for $x$ .

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Step 4.2.4.1

Set $8 x + 1$ equal to $0$ .

$8 x + 1 = 0$

Step 4.2.4.2

Solve $8 x + 1 = 0$ for $x$ .

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Step 4.2.4.2.1

Subtract $1$ from both sides of the equation.

$8 x = - 1$

Step 4.2.4.2.2

Divide each term in $8 x = - 1$ by $8$ and simplify.

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Step 4.2.4.2.2.1

Divide each term in $8 x = - 1$ by $8$ .

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 4.2.4.2.2.2

Simplify the left side.

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Step 4.2.4.2.2.2.1

Cancel the common factor of $8$ .

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Step 4.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 4.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{8}$

Step 4.2.4.2.2.3

Simplify the right side.

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Step 4.2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{8}$

Step 4.2.5

The final solution is all the values that make $(9 x - 1) (8 x + 1) = 0$ true.

$x = \frac{1}{9}, - \frac{1}{8}$

Step 5

The final solution is all the values that make $(72 x^{2} + x - 73) (72 x^{2} + x - 1) = 0$ true.

$x = 1, - \frac{73}{72}, \frac{1}{9}, - \frac{1}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 1, - \frac{73}{72}, \frac{1}{9}, - \frac{1}{8}$

Decimal Form:

$x = 1, - 1.013\overline{8}, 0.\overline{1}, - 0.125$

Mixed Number Form:

$x = 1, - 1 \frac{1}{72}, \frac{1}{9}, - \frac{1}{8}$